Static finite energy solutions of the field equations require
\(A_0=G_0=0\)  [1]}. Moreover, since we are interested in finding vortex solutions we will look for \(z\) -independent field configurations proposing the well-honored Nielsen-Olesen [2]} axially symmetric ansatz for fields in both sectors
\(&&\phi =f(r)e^{in\varphi }, \,\,\, A_\varphi =n\frac{\alpha (r)}{r}, \,\, A_r=A_z=0, \,\,\, n\in \mathbb {Z}, \\&&\eta =h(r)e^{ik\varphi }, \,\,\, G_\varphi =k\frac{ \beta (r)}{g_e r}, \,\, G_r=G_z=0, \,\,\, k\in \mathbb {Z}, \) 
